Head movement evaluations are important because disabled individuals may be able to operate communication aids using
head motion. A paper reported the accompanying data on neck rotation (in degrees) both in the clockwise direction (CL) and in the counterclockwise direction (CO) for 14 subjects. For purposes of this exercise, you can assume that the 14 subjects are representative of the population of adult Americans
Subject: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
CL: 57.1 35.7 54.5 56.8 51.1 70.8 77.3 51.6 54.7 63.6 59.2 59.2 55.8 38.2
CO: 44.6 52.1 60.2 52.7 47.2 65.6 71.4 48.8 53.1 66.3 59.8 47.5 64.5 34.7
Based on these data, is it reasonable to conclude that mean neck rotation is greater in the clockwise direction than in the counterclockwise direction? Carry out a hypothesis test using a significance level of πΌ = 0.01. (Use π1 for clockwise, π2 for counterclockwise, and πd = π1 β π2 .)
a) State the appropriate null and alternative hypotheses (in degrees).
b) Find the test statistic. (Round your answer to two decimal places.)
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a) The appropriate null and alternative hypotheses are as follows:
Null hypothesis (H0): The mean neck rotation in the clockwise direction is not greater than the mean neck rotation in the counterclockwise direction.
Alternative hypothesis (Ha): The mean neck rotation in the clockwise direction is greater than the mean neck rotation in the counterclockwise direction.
H0: π1 β€ π2
Ha: π1 > π2
b) To find the test statistic, we will use the formula for a paired t-test:
t = (πΜπ - ππ) / (ππ / βπ)
where πΜπ is the mean difference, ππ is the hypothesized mean difference, ππ is the standard deviation of the differences, and π is the sample size.
First, calculate the mean difference (πΜπ):
πΜπ = πΜπΆπΏ - πΜπΆπ
πΜπΆπΏ = (57.1 + 35.7 + 54.5 + 56.8 + 51.1 + 70.8 + 77.3 + 51.6 + 54.7 + 63.6 + 59.2 + 59.2 + 55.8 + 38.2) / 14 = 54.4
πΜπΆπ = (44.6 + 52.1 + 60.2 + 52.7 + 47.2 + 65.6 + 71.4 + 48.8 + 53.1 + 66.3 + 59.8 + 47.5 + 64.5 + 34.7) / 14 = 53.32
πΜπ = 54.4 - 53.32 = 1.08 (rounded to two decimal places)
Next, calculate the standard deviation of the differences (ππ):
ππ = β[(β(ππ - πΜπ)Β²) / (π - 1)]
(ππ - πΜπ)Β² for each subject:
(57.1 - 1.08)Β² = 3143.68
(35.7 - 1.08)Β² = 1228.62
(54.5 - 1.08)Β² = 4427.98
(56.8 - 1.08)Β² = 4620.32
(51.1 - 1.08)Β² = 2997.98
(70.8 - 1.08)Β² = 9312.42
(77.3 - 1.08)Β² = 13113.03
(51.6 - 1.08)Β² = 2439.67
(54.7 - 1.08)Β² = 4880.23
(63.6 - 1.08)Β² = 7788.62
(59.2 - 1.08)Β² = 6809.52
(59.2 - 1.08)Β² = 6809.52
(55.8 - 1.08)Β² = 4919.58
(38.2 - 1.08)Β² = 4106.42
Sum of (ππ - πΜπ)Β² = 76997.97
ππ = β(76997.97 / (14 - 1)) = β(76997.97 / 13) = 78.599 (rounded to three decimal places)
Finally, calculate the test statistic (t):
t = (πΜπ - ππ) / (ππ / βπ)
t = (1.08 - 0) / (78.599 / β14) = 0.136 (rounded to three decimal places)
The test statistic is 0.136.
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